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Name ________________________________________ Date __________________ Class__________________ LESSON 4-6 Reteach Triangle Congruence: ASA, AAS, and HL Angle-Side-Angle (ASA) Congruence Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. DF is the included side of �D and �F. AC is the included side of �A and �C. Determine whether you can use ASA to prove the triangles congruent. Explain. 1. UKLM and UNPQ 2. UEFG and UXYZ _________________________________________ ________________________________________ _________________________________________ ________________________________________ 3. UKLM and UPNM, given that M is the midpoint of NL 4. USTW and UUTV _________________________________________ ________________________________________ _________________________________________ ________________________________________ _________________________________________ ________________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 4-46 Holt McDougal Geometry Name ________________________________________ Date __________________ Class__________________ LESSON 4-6 Reteach Triangle Congruence: ASA, AAS, and HL continued Angle-Angle-Side (AAS) Congruence Theorem If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent. FH is a nonincluded side of �F and �G. JL is a nonincluded side of �J and �K. \ Special theorems can be used to prove right triangles congruent. Hypotenuse-Leg (HL) Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. 5. Describe the corresponding parts and the justifications for using them to prove the triangles congruent by AAS. Given: BD is the angle bisector of ∠ADC. Prove: UABD � UCBD _________________________________________________________________________________________ _________________________________________________________________________________________ Determine whether you can use the HL Congruence Theorem to prove the triangles congruent. If yes, explain. If not, tell what else you need to know. 6. UUVW � UWXU 7. UTSR � UPQR _________________________________________ ________________________________________ _________________________________________ ________________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 4-47 Holt McDougal Geometry 10. Statements 1. ∠IJK ≅ ∠LMN, ∠IKJ ≅ ∠LNM angles are right angles, and they are all congruent. GE is congruent to GE by the Reflexive Property, and thus the conditions for HL congruence between UFGE and UDGE have been met. Similar reasoning shows that UFGA is congruent to UBGA and that UBGC is congruent to UDGC. ∠EGD and ∠AGB are vertical angles, so they are congruent. This fact, and the congruencies already shown, meet the conditions for ASA congruence between UEGD and UBGA. Similar reasoning shows that UFGE is congruent to UBGC and UFGA is congruent to UDGC. The Transitive Property of Congruence can now be used to show that all the triangles within UACE are congruent. When triangles have been proven congruent, it is known that each matching part of the triangles is congruent. Hence AB = BC = CD = DE = EF = FA. By the Addition Property of Equality and the Segment Addition Postulate, AC = CE = EA and thus UACE is equilateral. Reasons 1. a. Given 2. JK ≅ MN 2. b. Definition of rectangle 3. UIJK ≅ ULMN 3. c. ASA Practice B 1. No; you need to know that AB ≅ CB. 2. 3. 4. 6. 8. Yes Yes, if you use Third ∠s Thm. first. 5. ASA or AAS HL none 7. AAS or ASA Possible answer: All right angles are congruent, so ∠QUR ≅ ∠SUR. ∠RQU and ∠PQU are supplementary and ∠RSU and ∠TSU are supplementary by the Linear Pair Theorem. But it is given that ∠PQU ≅ ∠TSU, so by the Congruent Supplements Theorem, ∠RQU ≅ ∠RSU. RU ≅ RU by the Reflexive Property of ≅, so URUQ ≅ URUS by AAS. Practice C Reteach 1. angle, one adjacent side, and the side opposite the angle, then you might not have enough information to draw the triangle; if the three parts are the three angles. 1. Yes; ∠K ≅ ∠N, KL ≅ NP, and ∠L ≅ ∠P as given. 2. No; you need to know that GF ≅ ZY . 3. No; you need to know that ∠NMP ≅ ∠LMK. 2. In right triangles, one of the three angles is known, and all right angles are congruent. This means that fewer parts must be shown congruent to prove two right triangles congruent. LL is a special case of SAS in which the A is the right angle. HA is a special case of AAS in which the first A is the right angle. LA is a special case of ASA in which one A is a right angle. 3. Yes; possible answer: If the two parts given are both angles, then the angle measures are set, but the side lengths are not. 4. Yes; ∠W ≅ ∠V, TW ≅ TV as given. ∠STW ≅ ∠UTV by the Vert. ∠s Thm. 5. ∠A ≅ ∠C (Given), ∠ADB ≅ ∠CDB (Def. of ∠ bisector), BD ≅ BD (Reflex. Prop. of ≅) 6. Yes; UV ≅ WX (Given) and UW ≅ UW (Reflex. Prop. of ≅) 7. No; you need to know that TR ≅ PR. Challenge 1. a. UABC ≅ UEDC b. Proofs may vary. 4. Possible answer: GB, GD and GF all have the same length, by the definition of the radius of a circle, and so they are all congruent. Each radius is perpendicular to its side of the triangle, so all those Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. A40 Holt McDougal Geometry